Tags: #projects/review #expository #geomtop

# What is the difference between low and high dimensional topology?

__High-dimensional topology__ refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. __Low-dimensional topology__ is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2.

The distinction is because __surgery theory__ works in dimension 5 and above (in fact, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work, and other phenomena occur.

The precise reason for the difference at dimension 5 is because the Whitney embedding theorem, the __key technical trick__ which underlies surgery theory, requires 2+1 dimensions.

Low-dimensional topology is strongly geometric, as reflected in the __uniformization__ theorem in 2 dimensions - every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic - and the __Geometrization__ theorem in 3 dimensions - every __3-manifold__ can be cut into pieces, each of which has one of 8 possible geometries.